Transcript Atomic Structure

Atomic Structure
Matter
Break
Break
Kanad, 600 BC
1 meter
Atom
10-10 meter
(1 angstrom)
• Elements are made of tiny particles called atoms.
• All atoms of a given element are identical.
• The atoms of a given element are different from those of any
other element.
John Dalton (1814) • Atoms of one element can combine with atoms of other elements
to form compounds. A given compound always has the same
relative numbers of types of atoms.
.
• Atoms cannot be created, divided into smaller particles,
or destroyed in the chemical process. A chemical reaction
simply changes the way atoms are grouped together.
Atoms are made up of 3 types of particles:
Electrons
Protons
(It is 1840 times heavier than an electron)
Neutrons
(Similar mass as that of a proton)
These particles have different properties.
Electrons are tiny, very light particles and have negative electrical charges (-).
Protons are much larger and heavier than electrons and have the opposite charges,
A proton has a positive (+) charge.
Neutrons are large and heavy like protons, however neutrons have no
electrical charge.
A Hydrogen Atom
Determination of the Charge on an Electron
J.J.Thomson
H.A. Wilson
J.S.E. Townsend
The charge on electron was first measured by J.J. Thomson and two co-workers (J.S.E.
Townsend and H.A. Wilson), starting in 1897. Each used a slightly different method.
Townsend's work depended on the fact that drops of water will grow around ions in
humid air. Under the influence of gravity, the drop would fall, accelerating until it hit a
constant speed.
He determined the e/m ratio of the droplets, then multiplied by the mass of one droplet to
get the value for e.
Thomson, Townsend, and Wilson each obtained roughly the same value for the charge
on positive and negative ions. It was about 1 x 10-19 coulombs. This work continued until
about 1901 or 1902.
Robert A. Millikan's Measurement
Robert A. Millikan started his work on electron charge in 1906 and continued for seven
years. His 1913 article announcing the determination of the electron's charge is a classic
and Millikan received the Nobel Prize for his efforts.
The actual apparatus used in
the Oil-Drop experiment by
Millikan
Some points about the experiment :
1. The two plates were 16 mm across, "correct to about .01 mm."
2. The hole bored in the top plate was very small.
3. The space between the plates was illuminated with a powerful beam of light.
4. He sprayed oil ("the highest grade of clock oil") with an atomizer that made drops one
ten-thousandth of an inch in diameter.
5. One drop of oil would make it through the hole.
6. The plates were charged with 5,000 volts.
7. It took a drop with no charge about 30 seconds to fall across the opening between the
plates.
8. He exposed the droplet to radiation while it was falling, which stripped electrons off.
9. The droplet would slow in its fall. The drops were too small to see. What he saw was a
shining point of light.
10. By adjusting the current, he could freeze the drop in place and hold it there for hours.
He could also make the drop move up and down many times.
11. Since the rate of ascent (or descent) was critical, he has a highly accurate scale
inscribed onto the telescope used for droplet observation and he used a highly accurate
clock, "which read to 0.002 second
Millikan's Improvements over Thomson
1. Oil evaporated much slower than water, so the drops stayed essentially
constant in mass.
2. Millikan could study one drop at a time, rather than a whole cloud.
3. In following the oil drop over many ascents and descents, he could measure the
drop as it lost or gained electrons, sometimes only one at a time. Every time
the drop gained or lost charge, it ALWAYS did so in a whole number multiple of
the same charge.
The value as of 1991 (for the charge on the electron) is 1.60217733 (49) x 10¯19
coulombs. This is less than 1% higher than the value obtained by Millikan in
1913. The 49 in parenthesis shows the plus/minus range of the last two digits
(the 33). It is unlikely that there will be much improvement of the accuracy in
years to come.
Interesting Fact about Robert Millikan's Experiment
In "The Discovery of Subatomic Particles" by Steven Weinberg there appears a
footnote on p. 97. It reads:
. . . . there appeared a remarkable posthumous memoir that throws some doubt on
Millikan's leading role in these experiments. Harvey Fletcher (1884-1981), who was
a graduate student at the University of Chicago, at Millikan's suggestion worked
on the measurement of electronic charge for his doctoral thesis, and co-authored
some of the early papers on this subject with Millikan. Fletcher left a manuscript
with a friend with instructions that it be published after his death; the manuscript
was published in Physics Today, June 1982, page 43. In it, Fletcher claims that he
was the first to do the experiment with oil drops, was the first to measure charges
on single droplets, and may have been the first to suggest the use of oil.
According to Fletcher, he had expected to be co-author with Millikan on the
crucial first article announcing the measurement of the electronic charge, but was
talked out of this by Millikan.
Electron Spin
Two types of experimental evidence which arose in the 1920s suggested an additional
property of the electron. One was the closely spaced splitting of the hydrogen spectral
lines, called fine structure. The other was the Stern-Gerlach experiment which showed
in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field
would be forced into two beams. Both of these experimental situations were consistent
with the possession of an intrinsic angular momentum and a magnetic moment by
individual electrons. Classically this could occur if the electron were a spinning ball of
charge, and this property was called electron spin.
An electron spin s = 1/2 is an intrinsic property of electrons.
Electrons have intrinsic angular momentum characterized by
quantum number 1/2. In the pattern of other quantized angular
momenta, this gives total angular momentum
The resulting fine structure which is observed corresponds to two
possibilities for the z-component of the angular momentum.
This causes an energy splitting because of the magnetic moment
of the electron.
STERN'S-GERLACH'S EXPERIMENT
A Helium Atom
Ions
• Ions are formed by addition or removal of electrons from neutral atoms.
• Cations (removal of electrons from atoms).
• Anions (addition of electrons to atoms).
H+ cation
H-atom
H- anion
Isotopes
• Two atoms with different numbers of neutrons are called isotopes.
• For example, an isotope of hydrogen exists in which the atom contains
1 neutron (commonly called deuterium).
Hydrogen
Atomic Mass = 1
Atomic Number = 1
Deuterium
Atomic Mass = 2
Atomic Number = 1
Since the atomic mass is the number of protons plus neutrons,
two isotopes of an element will have different atomic masses
(however the atomic number, Z, will remain the same).
Plum Pudding Model
By 1911 the components of the atom had been discovered. The atom consisted of
subatomic particles called protons and electrons. However, it was not clear how these
protons and electrons were arranged within the atom. J.J. Thomson suggested the
"plum pudding" model. In this model the electrons and protons are uniformly mixed
throughout the atom:
Rutherford's Planetary Model of the Atom
Rutherford tested Thomson's hypothesis by devising his "gold foil" experiment.
Rutherford reasoned that if Thomson's model was correct then the mass of the atom
was spread out throughout the atom. Then, if he shot high velocity alpha particles
(helium nuclei) at an atom then there would be very little to deflect the alpha particles.
He decided to test this with a thin film of gold atoms. As expected, most alpha particles
went right through the gold foil but to his amazement a few alpha particles rebounded
almost directly backwards.
These deflections were not consistent with Thomson's model. Rutherford was forced to
discard the Plum Pudding model and reasoned that the only way the alpha particles
could be deflected backwards was if most of the mass in an atom was concentrated in a
nucleus. He thus developed the planetary model of the atom which put all the protons
in the nucleus and the electrons orbited around the nucleus like planets around the sun.
Limitations of Rutherford Model
There appeared something terribly wrong with Rutherford's model of the atom.
The theory of electricity and magnetism predicted that opposite charges attract
each other and the electrons should gradually lose energy and spiral inward.
Moreover, physicists reasoned that the atoms should give off a rainbow of colors
as they do so. But no experiment could verify this rainbow.
In 1912 a Danish physicist, Niels Bohr came up with a theory that said the
electrons do not spiral into the nucleus and came up with some rules for what
does happen. (This began a new approach to science because for the first
time rules had to fit the observation regardless of how they conflicted with the
theories of the time.)
Atomic Spectra
• When one heats up a gas, it emits light of various wavelengths.
• The simplest spectrum is that for a Hydrogen atom.
• The spectrum of hydrogen is particularly important in astronomy
because most of the Universe is made of hydrogen.
• In 1885, Balmer discovered emission of H-atom in the visible region.
•The Balmer Series involves transitions starting (for absorption)
or ending (for emission) with the first excited state of hydrogen.
•Soon, the Lyman Series was discovered that involves transitions
which start or end with the ground state of hydrogen (UV region)
• By 1913, many more series were known: Paschen (Near Infrared),
Brackett (Far Infrared).
Rydberg proposed an
experimental data to fit this:
 = 1/λ= R (1/m2-1/n2)
R=Rydberg constant
(109677 cm-1)
m,n= integers
Series
m
Region
n
Lyman
UV
1
2,3,4,..
Balmer
Visible
2
3,4,5,…
Paschen
Near-IR
3
4,5,6,…
Brackett
Far-IR
4
5,6,7,…
The Bohr Model (1913)
1. The orbiting electrons existed in orbits that had discrete quantized energies.
That is, not every orbit is possible but only certain specific ones.
2. When electrons make the jump from one allowed orbit to another,
the energy difference is carried off (or supplied) by a single quantum of light
(called a photon) which has an energy equal to the energy difference between
the two orbitals.
3. The allowed orbits depend on quantized (discrete) values of orbital
angular momentum, (L) according to the equation:
n= principle quantum number, 1,2,3…
h=Planck’s constant
The energy of electron with a principle quantum number, n:
Now, we can derive the energy required for transition from an nth level
to the mth level as:
1/λ = meqe4/8ch3ε0(1/m2-1/n2)
Same as R=Rydberg constant
(109677 cm-1)
Atoms possess shells in which only a fixed number of electrons can
be accomodated:
K=2,L=8,M=8,N=8
Wave-Particle Duality
Wave-particle duality states that a particle such as an electron must also have
wave properties such as wavelength. In order to maintain a stable orbit, the
electron should have an integral number of wavelengths in its travels around the
nucleus. If the wavelengths do not match going around the circle, destructive
interference between the wavelengths causes the waves to disappear.
This observation led scientists to describe electron motion using equations for
wave motion.
Electron
+
de Broglie relationship (1924)
In 1924, Louis-Victor de Broglie formulated the de Broglie hypothesis,
claiming that all matter has a wave-like nature; he related wavelength,
λ (lambda), and momentum, p:
λ=h/p
What is the wavelength of an electron that has a velocity of 5.94×108 cm/sec
(electron accelerated through 100V) .
What is the wavelength of a man (70 Kg) walking at a velocity of 10km/hour.
Why don’t we have waves around us?
2005
Fullerene is the largest object known till
now that has an observable wavelength
(λ = 2.5 picometer)
C60
Uncertainty principle
(1927)
"The more precisely
the POSITION is determined,
the less precisely
the MOMENTUM is known"
The most common one is the uncertainty relation between position
and momentum of a particle in space:
Thus, for small particles like electrons or photons, it is not possible
to determine both the position and the momentum simultaneously
with the same accuracy.
This uncertainty leads to some strange effects. For example, in a
Quantum Mechanical world, I cannot predict where a particle will be with
100 % certainty. I can only speak in terms of probabilities. For example,
I can only say that an atom will be at some location with a 99 % probability,
and that there will be a 1 % probability it will be somewhere else (in fact, there
will be a small but finite probabilty that it can even be found across the Universe).
This is strange and very different from the macroscopic world that we live in.
Failure of the Bohr’s Model:
•Consider an electron which has a mass of 9.1× 10-31Kg
• Let the electron be moving in the 1st Bohr orbit (radius=0.52 Å) in the hydrogen atom.
• It will have a momentum of 2×10-21 Kg cm/sec.
• To know this momentum within 1% accuracy, the uncertainty in momentum has to be
smaller than 2×10-23 Kg cm/sec.
• Then Δx will be 330 Å !! This is ~300 times the diameter of the 1st Bohr radius.
• You cannot even say that the electron was within the atom at all !
Bohr’s shells become most probable position regions (orbitals) in Quantum Mechanics
Hydrogen Atom Potential
We will solve Schrodinger equation for an electron bound to an proton by
electromagnetic potential to make one hydrogen atom.
3-D Schrodinger equation : :
Where,
Here
is called Laplacian. Here tha Laplacian is in Cartesian coordinate. So,
Transformation from Cartesian to Spherical Harmonics
So, Laplacian in Spherical coordinates,
So, Schrodinger equation in Spherical coordinates,
Separation of Variables
Now we can solve each of these two equations separately
Separation of Angular part
Solution For
Part
Solution for
For the case of m = 0
Putting these values in the above equation,
part
Power Series Solution
Plug into,
To find,
P(x) diverges at x = 1
Legendre Polynomials
Associated Legendre Polynomials
Solution of Radial Part
Solution of these equations under the constraints placed on the wavefunction
leads to series solutions in the form of polynomials called the associated
Laguerre functions. In order to fit the physical boundary conditions, these
solutions contain a parameter n which can take only positive integer values; this
parameter is called the principal quantum number. The form of the radial solutions
is
Where Ln,l is the associated Laguerre function. The first few radial
wavefunctions R are shown as part of the hydrogen wavefunctions
The principal quantum number or total quantum number n arises from
the solution of the radial part of the Schrodinger equation for the
hydrogen atom. The bound state energies of the electron in the
hydrogen atom are given by
The normalized position wavefunctions, given in spherical
coordinates are:
Where,
And a0 is Bohr Radius
Are Generalized Laguerre Polynomyals
Yl,m( θ,φ )
Is a spherical harmonics
Few Laguerre Polynomials
n
0
1
2
3
4
5
6
Probability densities for the electron at different quantum numbers (l)
Electronic Configuration of atoms
The electron configuration is the arrangement of electrons in an atom. The
electrons occupy specific probability regions, who's shapes and electron capacity
are denoted by the letters s,p,d,f.
s-orbital
p-orbital
d-orbital
There are four quantum numbers:
1. Principal Quantum Number (n): This has range from 1 to n. This represents
the total energy of the system (Do you remember the Bohr’s energy expression).
2. Azimuthul Quantum Number (l): This has range from 0 to n-1. This is related
to the orbital angular momemtum.
3. Magnetic Quantum Number (m): This has a range from –l to +l. This determines
energy shift of an atomic orbital due to external magnetic field (Zeeman effect).
4. Spin Quantum Number (s): It takes values of +½ or -½ (sometimes called "up"
and "down").
The electron not only rotates around the nucleus also rotates (spins) around itself.
Origin of Magnetic Quantum Number
For l = 2
Space Quantisation of
angular momentum of
electron for l = 2
Precession of the angular
momentum vector about the axis, defined by a magnetic field.
Pauli exclusion principle
No two electrons in one atom can have the same set of these four quantum numbers
sublevel
orbital
maximum no. of
electrons
s
1
2
p
3
6
d
5
10
f
7
14
Start filling the
electrons
Some Examples
Element
No. of Electrons in Element
Electron Configuration
He
2
1s2
Li
3
1s22s1
Be
4
1s22s2
O
8
1s22s22p4
Cl
17
1s22s22p63s23p5
K
19
1s22s22p63s23p64s1
s-block atoms (have valence
electrons in s-orbitals)
p-block atoms (have valence
electrons in p-orbitals)
d-block atoms (have valence
electrons in d-orbitals)
f-block atoms (have valence
electrons in f-orbitals)
Exceptions:
d subshell that is half-filled or full (ie 5 or 10 electrons) is more stable than
the s subshell of the next shell.
instance, copper (atomic number 29) has a configuration of [Ar]4s1 3d10,
not [Ar]4s2 3d9 as one would expect by the Aufbau principle.
Likewise, chromium (atomic number 24) has a configuration of [Ar]4s1 3d5,
not [Ar]4s2 3d4.
Element
Z
Electron configuration
Short electron conf.
Scandium
21 1s2 2s2 2p6 3s2 3p6 4s2 3d1
[Ar] 4s2 3d1
Titanium
22 1s2 2s2 2p6 3s2 3p6 4s2 3d2
[Ar] 4s2 3d2
Vanadium
23 1s2 2s2 2p6 3s2 3p6 4s2 3d3
[Ar] 4s2 3d3
Chromium
24 1s2 2s2 2p6 3s2 3p6 4s1 3d5
[Ar] 4s1 3d5
Manganese 25 1s2 2s2 2p6 3s2 3p6 4s2 3d5
[Ar] 4s2 3d5
Iron
26 1s2 2s2 2p6 3s2 3p6 4s2 3d6
[Ar] 4s2 3d6
Cobalt
27 1s2 2s2 2p6 3s2 3p6 4s2 3d7
[Ar] 4s2 3d7
Nickel
28 1s2 2s2 2p6 3s2 3p6 4s2 3d8
[Ar] 4s2 3d8
Copper
29 1s2 2s2 2p6 3s2 3p6 4s1 3d10
[Ar] 4s1 3d10
Zinc
30 1s2 2s2 2p6 3s2 3p6 4s2 3d10
[Ar] 4s2 3d10
Gallium
31 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1 [Ar] 4s2 3d10 4p1